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Theorem hbimtg 25187
Description: A more general and closed form of hbim 1826. (Contributed by Scott Fenton, 13-Dec-2010.)
Assertion
Ref Expression
hbimtg  |-  ( ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  (
( ch  ->  ps )  ->  A. x ( ph  ->  th ) ) )

Proof of Theorem hbimtg
StepHypRef Expression
1 hbntg 25186 . . . 4  |-  ( A. x ( ph  ->  A. x ch )  -> 
( -.  ch  ->  A. x  -.  ph )
)
2 pm2.21 102 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  th ) )
32alimi 1565 . . . 4  |-  ( A. x  -.  ph  ->  A. x
( ph  ->  th )
)
41, 3syl6 31 . . 3  |-  ( A. x ( ph  ->  A. x ch )  -> 
( -.  ch  ->  A. x ( ph  ->  th ) ) )
54adantr 452 . 2  |-  ( ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  ( -.  ch  ->  A. x
( ph  ->  th )
) )
6 ax-1 5 . . . . 5  |-  ( th 
->  ( ph  ->  th )
)
76alimi 1565 . . . 4  |-  ( A. x th  ->  A. x
( ph  ->  th )
)
87imim2i 14 . . 3  |-  ( ( ps  ->  A. x th )  ->  ( ps 
->  A. x ( ph  ->  th ) ) )
98adantl 453 . 2  |-  ( ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  ( ps  ->  A. x ( ph  ->  th ) ) )
105, 9jad 156 1  |-  ( ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  (
( ch  ->  ps )  ->  A. x ( ph  ->  th ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1546
This theorem is referenced by:  hbimg  25190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548
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